Find the least common multiple $(\text{LCM})$ of $11z^3+110z^2+99z$ and $14z^2+112z-126$. You can give your answer in its factored form.
Explanation: The least common multiple $(\text{LCM})$ of two polynomial expressions is the polynomial with the least number of factors that is divisible by both polynomials. [How does this relate to the least common multiple of integers?] We can find the $\text{LCM}$ by factoring the two polynomials as much as possible and then comparing the factors: $11z^3+110z^2+99z$ can be factored as ${(11)(z)}{(z+9)}{(z+1)}$ by factoring out a $11z$ and using the sum-product pattern. $14z^2+112z-126$ can be factored as ${(2)(7)}{(z+9)}{(z-1)}$ by factoring out a $14$ and using the sum-product pattern. We can see that: Both polynomials share the factors ${(z+9)}$ Only the first polynomial has the factors ${(11)(z)(z+1)}$ Only the second polynomial has the factors ${(2)(7)(z-1)}$ Therefore, the least common multiple is the product of all the above factors: [Why?] $\begin{aligned}&\phantom{=}{(z+9)}{(11)(z)(z+1)}{(2)(7)(z-1)}\\\\ &=154(z)(z-1)(z+1)(z+9)\end{aligned}$ In conclusion, the least common multiple of the two polynomials is $154(z)(z-1)(z+1)(z+9)$.